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Theorem psubclssatN 37253
 Description: A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
psubclssat.a 𝐴 = (Atoms‘𝐾)
psubclssat.c 𝐶 = (PSubCl‘𝐾)
Assertion
Ref Expression
psubclssatN ((𝐾𝐷𝑋𝐶) → 𝑋𝐴)

Proof of Theorem psubclssatN
StepHypRef Expression
1 psubclssat.a . . 3 𝐴 = (Atoms‘𝐾)
2 eqid 2798 . . 3 (⊥𝑃𝐾) = (⊥𝑃𝐾)
3 psubclssat.c . . 3 𝐶 = (PSubCl‘𝐾)
41, 2, 3psubcliN 37250 . 2 ((𝐾𝐷𝑋𝐶) → (𝑋𝐴 ∧ ((⊥𝑃𝐾)‘((⊥𝑃𝐾)‘𝑋)) = 𝑋))
54simpld 498 1 ((𝐾𝐷𝑋𝐶) → 𝑋𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881  ‘cfv 6324  Atomscatm 36575  ⊥𝑃cpolN 37214  PSubClcpscN 37246 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-psubclN 37247 This theorem is referenced by:  pmapidclN  37254  psubclinN  37260  paddatclN  37261  pclfinclN  37262  poml6N  37267  osumcllem3N  37270  osumcllem9N  37276  osumcllem11N  37278  osumclN  37279
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